Linear Code/Examples/Non-Linear Code

Examples of Non-Linear Codes

Subset of $\map V {6, 2}$ which is not a Linear Code

Let $\map V {6, 2}$ denote the set of sequences of length $6$ of elements of $\Z_2$.

For all $x \in \Z_p$, let $\overline x$ denote $1 + x \pmod 2$ so that:

$\overline 0 = 1$

$\overline 1 = 0$

Let $C \subseteq \map V {6, 2}$ be the set of elements of $\map V {6, 2}$ of the form:

$x y z \overline x \overline y \overline z$

Then $C$ is not a linear code.